What Lie group structure is used for infinite-dimensional Unitary Groups in Quantum Mechanics? Given an infinite-dimensional Hilbert space $H$, the set $U(H)$ of all unitary operators on $H$ forms a group, known as the unitary group.  Now several subgroups of this group play an important role in quantum mechanics, for instance the group of time-translation operators and the group of spatial translation operators.  And these subgroups are treated as Lie groups, which I assume means that $U(H)$ is also a Lie group.  So my question is, what is the differentiable manifold structure on $U(H)$ used in quantum mechanics?
And what is the topology used in quantum mechanics for this group?  Is it the uniform (norm) topology, or the strong operator topology, or what?
EDIT: Assuming the strong operator topology is the right one for quantum mechanics, the second page of this paper mentions "the Frechet Lie group $U(H)$ consisting of all unitary operators on H, equipped with the strong operator topology".  That means $U(H)$ has a Frechet manifold structure, i.e. it’s locally isomorphic to an infinite-dimensional Frechet space. (As opposed to an ordinary manifold which is locally isomorphic to a finite-dimensional Euclidean space.) But what is that Frechet manifold structure?
 A: $U(H)$ is a connected and continuous-path connected topological group with respect to the strong operator topology, which is the most natural one for physical applications. This topology permits to extend some features of Lie groups to the case of an "infinite-dimensional" group where the differentiable structure is not easy to define (there are different points of view and not all properties of standard finite-dimensional smooth manifolds can be extended). 
Lie groups have the property that they are almost completely defined by their Lie algebra (it is true for simply connected Lie groups, otherwise this property is valid in a neighborhood of the neutral element only). 
In particular, if the group is connected, every element is obtained as a product of elements belonging to one-parameter subgroup generated by vectors in the Lie algebra. 
This also happens for $U(H)$  in view of Stone theorem (which also uses the strong operator topology) even if no differentiable structure is chosen over $U(H)$. From an abstract point of view, we can think of selfadjoint operators, as the elements of the Lie algebra of $U(H)$. From the spectral theory of normal operators and the functional calculus, every unitary operator $U$ can be written  as  $e^{itA}$ for some selfadjoint operator $A$ and some real $t$.   
A global definition of a general Lie algebra for $U(H)$ is difficult since selfadjoint operators are unbounded and have different domains so that $[A,B]$ is generally undefined. 
